Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 9: do the squares 25, 144 and 169 satisfy a² + b² = c²?
Answer
Yes, because 25 + 144 = 169. - From Year 9: find the midpoint of (0, 4) and (6, 8).
Answer
(3, 6). - From Year 9: how many sides of an isosceles triangle are equal?
Answer
At least two.
Words used in this lesson
- isosceles
- a triangle with two equal sides. For example: XY = XZ.
- right-angled
- having one angle of 90 degrees. For example: the corner of a page.
- parallelogram
- a quadrilateral with two pairs of parallel sides. For example: a rectangle pushed over sideways, so its corners are no longer right angles.
- bisect
- cut into two equal parts. For example: the midpoint bisects AB.
- equidistant
- the same distance from two points. For example: same distance from A and B.
- converse
- the statement turned around. For example: if a² + b² = c² then right-angled.
It looks isosceles, so is it?
A surveyor in Chile marks three posts on a map grid, in metres: A at (2, 1), B at (8, 4) and C at (5, 7). The plan says the three posts make an isosceles triangle. On a drawing it looks about right, but a drawing cannot be trusted to a few millimetres.
Predict first: are any two of the three sides equal? Say which two, or say none, and give a reason. Then use the figure: it does the same test for a triangle on a fixed base.
C = (−1, 2): AC² = 13, BC² = 29 and AB² = 64. No two sides are equal. The two smaller squares add to 42, not 64, so it is not right-angled.
Show me the steps
- C = (−1, 2): AC² = 13, BC² = 29, AB² = 64. No two are equal, and 13 + 29 = 42 is not 64.
- C = (0, 5): AC² = 16 + 25 = 41 and BC² = 41, so AC = BC and it is isosceles. 41 + 41 = 82 is not 64, so no right angle.
- C = (0, 4): AC² = 16 + 16 = 32 and BC² = 32. 32 + 32 = 64 = AB², so it is isosceles and has a right angle at C.
- C = (4, 3): AC² = 64 + 9 = 73, BC² = 9 and AB² = 64. 64 + 9 = 73, so the right angle is at B, and no two sides are equal.
Turn the picture into numbers. Compare squared lengths: two sides are equal when their squares are equal, and a triangle has a right angle when the two smaller squares add up to the largest (the converse of Pythagoras). Two midpoints that are the same point show that two segments bisect each other. Two gradients that are equal show parallel sides, and a product of −1 shows perpendicular sides.
A proof ends with a sentence that says what has been shown, using the numbers, and the sentence must be about the shape in question, not about how it looks.
Two common mistakes. Adding the wrong squares: for a right angle the two smaller squares add to the largest, so 10 + 40 = 50, not 40 + 50 = 90. And trusting a rounded decimal: equal squared lengths are exact, so compare them instead.
Worked examples
1. Show that A(0, 0), B(6, 2) and C(−1, 3) make a right-angled triangle.
- AB² = 6² + 2² = 40. AC² = 1² + 3² = 10. BC² = 7² + 1² = 50.
- 10 + 40 = 50 = BC², so the triangle is right-angled at A, the vertex opposite BC.
2. Show that A(1, 2), B(5, 3), C(7, 8) and D(3, 7) make a parallelogram.
- Midpoint of AC: ((1 + 7) ÷ 2, (2 + 8) ÷ 2) = (4, 5). Midpoint of BD: ((5 + 3) ÷ 2, (3 + 7) ÷ 2) = (4, 5).
- The diagonals bisect each other at the same point, so ABCD is a parallelogram.
3. Find the point P on the x-axis that is equally far from A(0, 1) and B(4, 5).
- Let P = (x, 0). PA² = x² + 1 and PB² = (x − 4)² + 25 = x² − 8x + 41.
- Set them equal: x² + 1 = x² − 8x + 41, so 8x = 40 and x = 5. P = (5, 0), and PA² = PB² = 26.
Your go. The first step is given. Show that P(1, 0), Q(5, 0) and R(3, 4) make an isosceles triangle. The first side is PQ² = 4² + 0² = 16.
- PQ² = 16. Now find PR² and QR².
- Compare the three squares and say which two are equal.
Check the answer
PR² = 2² + 4² = 20 and QR² = 2² + 4² = 20. PR² = QR², so PR = QR and the triangle is isosceles. (PQ² = 16 is different, so it is not equilateral.)
With a GDC (optional)
A GDC can give the decimal size of a side, for example √57 is about 7.55. But compare squared lengths, not rounded decimals: √56 and √57 both round to 7.5 to 1 decimal place, so two sides can look equal when their squares, 56 and 57, are not.
Exact squared lengths are whole numbers here, so equal sides give exactly equal squares. Everything can be done by hand.
Check before you go on
1. Three squared side lengths of a triangle are given. Which set shows that the triangle is right-angled?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. P is (2, −1) and Q is (5, 6). What is PQ²?
3. A is (0, 0), B is (3, 4) and C is (−4, 3). What is BC²?
Level 2 · Building
4. ABCD is a parallelogram with A(−1, 2), B(4, 1) and C(7, 5). The diagonals AC and BD have the same midpoint. What is the x-coordinate of D?
5. Find the x-coordinate of the point on the x-axis that is the same distance from A(0, 2) and B(6, 4).
6. A is (0, 0), B is (8, 4) and C is (−2, k). The angle at A is a right angle. What is k?
Level 3 · Stretch
7. A(1, 1), B(5, 2), C(4, 6) and D(0, 5) are the corners of a square. What is AC², the squared length of a diagonal?
8. Two radio masts on a flat plain in Mongolia are at M(0, 1) and N(8, 5) on a grid in kilometres. A relay station R is on the y-axis at (0, y) and is equally far from both masts. What is y?
Something to think about
Worked example 2 shows that the diagonals of one quadrilateral ABCD share a midpoint. Does that prove that the diagonals of every parallelogram share a midpoint? Why or why not?
One way to think about it
No. One example shows it for those four points. To prove it for every parallelogram, use letters: O(0, 0), P(a, 0), R(b, c) and Q(a + b, c). The midpoint of OQ is ((a + b) ÷ 2, c ÷ 2), and the midpoint of PR is ((a + b) ÷ 2, c ÷ 2): the same point, whatever a, b and c are.
Exit ticket
9. D is (−1, 2), E is (5, 2) and F is (2, 6). Which statement about triangle DEF is correct?